✅ Pehle Yeh Aata Hai?
Yeh post shuru karne se pehle, in teen sawaalon ko 1 minute mein khud try kijiye:
- 32 × 5 = ?
- 52 × 3 = ?
- Do-digit number “qp” ka value?
Answers dekho
- 160
- 156
- 10q + p
Teeno sahi? Badhiya! Place value (10q + p) is post ki chaabi hai.
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🎬 Kahani: Teen Patte, Ek Sawaal
Golu ke paas teen number wale patte: 2, 3, 5. Khel: inhe □□ × □ mein lagao, jiska guna sabse bada, woh jeeta!
Golu ne socha: “Sabse bada do-digit number 53, to 53 × 2!” = 106. Priya ne 32 × 5 = 160 bana diya aur jeet gayi! 😮
Sabse bada number banana zaroori nahi tha? Chalo pakka niyam dhoondhte hain.
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📖 Concept: Sabse Bada Digit Akela Guna Kare
Step 1: Saare 6 tareeke
| Guna | Jawab |
|---|---|
| 23 × 5 | 115 |
| 25 × 3 | 75 |
| 32 × 5 | 160 |
| 35 × 2 | 70 |
| 52 × 3 | 156 |
| 53 × 2 | 106 |
Sabse bada: 32 × 5 = 160.
Step 2: Niyam
Digits p < q < r: sabse bada guna = (qp) × r. Yaani sabse bada digit akela guna karne wala, baaki do ghatte kram mein do-digit number.
Step 3: Kyun? Algebra se
qp × r = (10q + p) × r = 10qr + pr. Rp × q = (10r + p) × q = 10qr + pq. Farak = pr − pq = p(r − q) > 0, kyunki r > q. To qp × r bada ✅
Aur 52 × 3 (rq × p jaisa nahi, rp × q) se bhi upar dekha. Baaki tareeke isse bhi chhote.
WHY intuition? Akela digit poore do-digit number ko guna karta hai, to uska bada hona sabse zyada asar karta.
Step 4: Udaharan
1, 3, 7: 31 × 7 = 217. 3, 5, 9: 53 × 9 = 477.
Step 5: Sabse chhota guna?
Ulta socho: sabse chhota digit akela, baaki badhte kram mein: 2, 3, 5 → 35 × 2 = 70 (table mein sabse chhota ✅).
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🔐 Deep Concept: Do Numbers Ka Jod Fixed, Guna Kab Sabse Bada?
32 aur 5 mein farak bada hai, phir bhi jeeta! Asal mein qp × r mein “10” ke saath q aur r dono guna hote (10qr), jo sabse bada hissa hai. Isliye q aur r (do bade digits) ko alag-alag numbers mein daalna zaroori hai.
General sabak: jod same ho to guna tab sabse bada jab numbers paas-paas hon (jaise 5 × 5 > 9 × 1). 🔐
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✏️ Solved Examples
Example 1 (Easy): 1, 3, 7 se sabse bada guna?
31 × 7 = 217.
WHY sahi hai? 73 × 1 = 73, 71 × 3 = 213: 217 sabse bada ✅
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Example 2 (Medium): 3, 5, 9?
53 × 9 = 477.
WHY sahi hai? 93 × 5 = 465 < 477 ✅
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Example 3 (NCERT style): Dikhao qp × r > rp × q.
Farak = p(r − q) > 0.
WHY sahi hai? 2, 3, 5: 160 − 156 = 4 = 2 × (5 − 3) ✅
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Priya ka mantra: “Bada digit akela, baaki ulta kram mein!” 🎉
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🏠 Real Life Mein
- 🎲 Card games aur number puzzles.
- 📦 Packing: kaun si lambaai-chaudaai se zyada area (Ch 7 se link).
- 🧮 Olympiad mein aise sawaal.
- 💼 Business: daam × sankhya ka sabse achha mel.
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🧭 Kab Use Karein, Kab Nahi
| Situation | Kya karein | Kyun |
|---|---|---|
| □□ × □ sabse bada | ✅ (qp) × r | Bada digit akela |
| Sabse chhota | ✅ (qr) × p jaisa ulta | Chhota digit akela |
| Sabse bada do-digit banao | ❌ Zaroori nahi | 53 × 2 haara |
| Pakka karna | ✅ Algebra farak | p(r − q) |
— 📌 charumam.com
❌→✅ Worked Mistake
Sawaal: 4, 6, 8 se sabse bada □□ × □?
❌ Galat: “86 × 4 = 344.”
Galti kya hai? Sabse bada digit akela hona chahiye ❌
✅ Sahi: 64 × 8 = 512.
Yaad rakho: r akela.
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📝 10 MCQs
Q1. 2, 3, 5 sabse bada?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 160 ✅
- (b) galat: 106
- (c) galat: 156
- (d) galat: 70
Q2. 1, 3, 7?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 217 ✅
- (b) galat: 73
- (c) galat: 213
- (d) galat: 37
Q3. 3, 5, 9?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 477 ✅
- (b) galat: 285
- (c) galat: 465
- (d) galat: 177
Q4. Farak qp × r − rp × q?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi ✅
- (b) galat: Nahi.
- (c) galat: Nahi.
- (d) galat: Nahi.
Q5. 2, 3, 5 sabse chhota?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 70 ✅
- (b) galat: 115
- (c) galat: 75
- (d) galat: 106
Q6. 4, 6, 8 sabse bada?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 512 ✅
- (b) galat: 344
- (c) galat: 504
- (d) galat: 272
Q7. qp ka value?
Solution dekho
Sahi jawab: (a)
- (a) sahi: Sahi ✅
- (b) galat: Ulta.
- (c) galat: Nahi.
- (d) galat: Nahi.
Q8. Kul kitne tareeke (3 alag digits)?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 3 × 2 × 1 ✅
- (b) galat: Nahi.
- (c) galat: Nahi.
- (d) galat: Nahi.
Q9. Jod 10, guna sabse bada?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 25 ✅
- (b) galat: 9
- (c) galat: 16
- (d) galat: 21
Q10. 1, 2, 9?
Solution dekho
Sahi jawab: (a)
- (a) sahi: 189 ✅
- (b) galat: 92
- (c) galat: 182
- (d) galat: 29
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❓ FAQs (Bachchon Ke Asli Sawaal)
1. Kya zero digit ho?
Ho sakta; phir bhi niyam lagaao, par 0 akela mat rakho.
2. Do digits same?
Niyam chalta; farak 0 bhi aa sakta.
3. □□ × □□ (4 digits)?
Alag niyam: bade digits dono numbers ke dahaai mein.
4. Kyun sab tareeke gine?
Shuruaat mein samajhne ke liye; phir algebra se pakka.
5. p(r − q) ka matlab?
Dono gunon ka farak.
6. Sabse chhote ka niyam?
Sabse chhota digit akela, baaki badhte kram mein.
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🏋️ Practice Zone (12 Sawaal)
Q1. Easy 1, 2, 3 sabse bada?
Solution dekho
21 × 3 = 63.
Q2. Easy 2, 4, 6?
Solution dekho
42 × 6 = 252.
Q3. Easy 5, 7, 8?
Solution dekho
75 × 8 = 600.
Q4. Easy 1, 5, 9?
Solution dekho
51 × 9 = 459.
Q5. Medium 2, 3, 5 sabse chhota?
Solution dekho
35 × 2 = 70.
Q6. Medium 3, 4, 9 ke saare 6 guna.
Solution dekho
34×9=306, 43×9=387, 39×4=156, 93×4=372, 49×3=147, 94×3=282.
Q7. Medium Unme sabse bada?
Solution dekho
43 × 9 = 387.
Q8. NCERT style 1, 3, 7 sabse bada.
Solution dekho
217.
Q9. NCERT style 3, 5, 9 sabse bada.
Solution dekho
477.
Q10. Medium Farak 32 × 5 − 52 × 3?
Solution dekho
4.
Q11. Medium p(r − q) se check karo (2, 3, 5).
Solution dekho
2 × 2 = 4 ✅
Q12. NCERT style Kyun q aur r alag numbers mein?
Solution dekho
10qr wala hissa sabse bada banta.
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🧠 HOTS (Sochne Wale Sawaal)
Q1. □□ × □□ digits 1, 2, 3, 4 se sabse bada?
Solution dekho
41 × 32 = 1312 (ya 42 × 31 = 1302). Sabse bada 41 × 32 = 1312.
Q2. □□□ × □ (digits 2, 3, 5, 7)?
Solution dekho
Bada digit akela: 532 × 7 = 3724.
Q3. Kya kabhi qp × r aur rp × q barabar?
Solution dekho
Jab p = 0 ya q = r.
Q4. Sabse chhota aur sabse bada ka farak (2, 3, 5)?
Solution dekho
160 − 70 = 90.
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🏆 Olympiad Corner
Q1. Digits 1–4, □□ × □□ sabse chhota?
Solution dekho
13 × 24 = 312 ya 14 × 23 = 322 → 13 × 24 = 312.
Q2. 5, 6, 7, 8 se □□ × □□ sabse bada?
Solution dekho
85 × 76 = 6460 vs 86 × 75 = 6450 → 6460.
Q3. Digits a < b < c, guna (ba) × c − (ca) × b?
Solution dekho
a(c − b).
Q4. Kitne alag guna (1, 2, 3)?
Solution dekho
6 tareeke: 12×3=36, 13×2=26, 21×3=63, 23×1=23, 31×2=62, 32×1=32 → 6 alag.
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💗 CharuMam Motivation
Golu ne sabse bada number banaaya par haar gaya; Priya ne socha, aur jeeti. ✖️
Sabse bada dikhna zaroori nahi, sahi jagah pe hona zaroori hai. Apni taakat sahi jagah lagao! 💪
— 📌 charumam.com
🟢 Master Recap
- p < q < r: sabse bada (qp) × r.
- Saboot: farak = p(r − q) > 0.
- Sabse chhota: chhota digit akela, baaki badhte kram.
- 2, 3, 5 → 160; 1, 3, 7 → 217; 3, 5, 9 → 477.
- Algebra se “har case” pakka.
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👉 Aage Kya?
74 − 47 = 27, 9 se bat jaata hai. Hamesha? Agle post mein divisibility tricks ka raaz! 🔍
Koi doubt? CharuMam AI Tutor se poochho, ya comment mein likho! 💬